240 Part 3: The Shape of the World
CHECK POINT
Find the length of the specified segment.
16. 'GHI~'ARM, GH = 9 ft, GI = 8 ft, AR = 12 ft. Find AM.- 'JKL~'DOG, JK = 17 m, JL = 25 m, DG = 30 m. Find DO.
- 'ABC~'XYZ, AB = 21 cm, BC = 54 cm, XY = 7 cm. Find YZ.
- 'DEF~'CAT, DE = 65 in, EF = 45 in, CA = 13 in. Find AT.
- 'MNO~'LEG, MN = x – 3, NO = 3, EG = 21, LE = 2x + 4. Find LE.
Indirect Measurement with Trigonometry ..............................................................
Trigonometry, or “triangle measurement,” developed as a means to calculate lengths that can’t be
measured directly. It accomplishes this by using the relationships of sides of right triangles. These
fundamental relationships of trigonometry are based on the proportions of similar triangles.Trigonometric Ratios
If you look at two right triangles, each with an acute angle of 25r, you can quickly prove that the
two triangles are similar. The 25r angles are congruent and the right angles are congruent, so the
triangles are similar. In fact, all right triangles containing an angle of 25r are similar, and you can
think of them as a family. For any right triangle in this family, the ratio of the side opposite the
25 r angle to the hypotenuse will always be the same.
That’s not the ratio you usually think about with similar triangles, but it follows from the usual
one. Most of the time, you’d say that the ratio of the side opposite the 25r angle in one triangle
to the side opposite the 25r angle in the other is the same as the ratio of the hypotenuse to the
hypotenuse. Let’s write it this way:
opposite1
opposite2hypotenuse1
hypotenuse2. If you cross-multiply, divide both
sides by hypotenuse1, and then divide both sides by hypotenuse2, here’s what happens.